We generalize results on quasi-invariant states from compact to discrete amenable group actions on unital C*-algebras. Assuming a faithful invariant tracial state, we use Folner averages to construct a conditional expectation onto the fixed-point algebra. When the second cohomology group H-2(G, T) = 0, we show that every quasi-invariant state with full central support has a GNS representation unitarily equivalent to that of an invariant state obtained by averaging. This provides a complete discrete analogue of the compact case.
This paper shows how a classical inequality in inner product spaces for any orthogonal projection by Horváth 3 can serve as a unified tool to derive recent statistical inequalities from Refs. 4 and 6.
In this paper, we study Green’s relations on the state space of a certain class of compact quantum semigroups, namely non-commutative unitary C*-algebras. Furthermore, we generalize Friedberg’s results on finite-dimensional representations of group-extremal affine semigroups to the state space of noncommutative unitary C*-algebras. This yields new examples of compact semigroups that admit a well-behaved finite-dimensional representation theory beyond the group-extremal setting.
In this paper, we study the existence and blow-up behavior to the following stochastic non-local reaction-diffusion equation: du(t, x)=[(Delta +gamma)u(t, x)+integral(D )u(q)(t, y)dy - ku(p)(t, x)+delta u(m)(t, x)integral(D )u(n)(t, y)dy]dt+eta u(t, x)dB(H)(t), u(t, x)=0,t>0,x is an element of partial derivative D, u(0,x)=f(x)>= 0, x is an element of D, where D subset of R-d (d >= 1) is a bounded domain with smooth boundary partial derivative D. Here, k > 0, gamma,delta,eta >= 0 and p,q,n > 1, m >= 0 with m + n >= q >= p. The initial data f is a non-negative bounded measurable function in class C-2 that is not identically zero. Here, {BH(t)} t >= 0 is a one-dimensional fractional Brownian motion with Hurst parameter 1/2 <= H < 1 defined on a filtered probability space (Omega,F, (F-t)(t >= 0), P) satisfying the usual conditions. First, we estimate a lower bound for the finite-time blow-up, and by choosing suitable initial data, we obtain an upper bound for the finite-time blow-up of the above equation. Next, we provide a sufficient condition for the global existence of a weak solution of the above equation. Furthermore, we obtain the bounds for the probability of a blow-up solution.
In classical probability theory, selfsimilar stable processes were well studied in the 1980s. Among others, the so-called linear fractional stable motion is a typical example of selfsimilar stable processes, including the well-known fractional Brownian motion as a special case. In this paper, the linear fractional stable motion and selfsimilar stable processes with stationary increments are studied within the framework of free probability.
For alpha > 0 and sigma > 0, we consider the following probability distribution on alpha N-0 : pi(alpha, sigma )= exp(- sigma/alpha(2))& sum;(infinity)(n=0 )1/n! (sigma/alpha(2))(n )delta(alpha n), where delta(y) denotes the Dirac measure with mass at y. For alpha = 1, pi(1 ,sigma) is the Poisson distribution with parameter sigma. Furthermore, the centered probability distribution pi(alpha ,sigma )= exp (-sigma/alpha(2)) & sum;(infinity)(n=0 )1/n!(sigma/alpha(2))(n )delta(alpha n-sigma/alpha) weakly converges to mu sigma as alpha -> 0. Here mu(sigma) is the Gaussian distribution with mean zero and variance sigma. Let (c(n))(infinity)(n = 0) be the monic polynomial sequence that is orthogonal with respect to the measure mu alpha,sigma. In particular, for alpha = 1, (c(n))(infinity)(n=0) is a sequence of Charlier polynomials. Let F-sigma (C) denote the Bargmann space of all entire functions f(z) = & sum;(infinity)(n=0 )f(n)z(n) with f(n )is an element of C satisfying & sum;(infinity)(n=0 )|f(n)|(2 )n! sigma(n )< infinity. The generalized Segal-Bargmann transform associated with the measure pi alpha,sigma is a unitary operator S : L-2(alpha N-0, pi(alpha,sigma)) -> F-sigma(C) that satisfies (Sc-n) (z) = z(n) for n is an element of N-0. We present some new results related to the operator S. In particular, we observe how the study of S naturally leads to the normal ordering in the Weyl algebra.
In this paper, we establish that the mixed q-deformed Araki-Woods von Neumann algebra Gamma(T)(H-& Ropf;,U-t)'', as introduced in Ref. 4, possesses a trivial bicentralizer whenever it is of type III1.
This paper establishes the large deviation principle (LDP) for the two-dimensional stochastic primitive equations with horizontal viscosity, a model for anisotropic geophysical flows under small noise. Using the weak convergence method, we first prove an LDP in the space C([0,T],H-1). The main novelty lies in demonstrating that the solutions possess enhanced pathwise regularity, belonging almost surely to C([0,T],H). This stronger regularity allows us to subsequently strengthen the LDP to the finer topology of C([0,T],H), providing a more precise description of the asymptotic behavior. The analysis requires handling the intrinsic anisotropy of the system, particularly in the convection term, which introduces technical challenges distinct from the anisotropic Navier-Stokes cases.
Under mild assumptions, we establish that the (uniform) exponential quasi-mixing limits of the underlying Markov process can be guaranteed by (intrinsic) ultracontractivity of its transition semigroup and dual semigroup. Other interesting properties are also considered, including (uniform) mean-ratio quasi-ergodicity proportional to time, and the conditional process converging (uniformly) exponentially to a (uniform) exponential ergodic process. It is worth emphasizing that the reference measure in this paper is permitted to be infinite. Finally, two examples are provided.
The infinite-dimensional hypercube (IDH) is the set Gamma of all finite subsets of & Nopf; (the nonnegative integer set) as a graph, where two sets are adjacent if they differ only by precisely one nonnegative integer. In this paper, we investigate the IDH from a perspective of continuous-time Markov processes. First, we introduce a weighted graph Laplacian Delta(w) on Gamma and prove some technical theorems concerning Delta(w) . Then we consider the point evaluation process X on Gamma and prove that X is a continuous-time Markov process with Delta(w) as its generator. We also construct a class of martingales in terms of X and Delta(w) and obtain some relevant results. Finally, we construct a Q-process Y on Gamma through a Q-matrix associated with Delta(w) and verify its Markov property as well as other properties. Some other related results are also obtained in this paper.
Based on white noise theory, we first discuss the Gaussian generalized Mehler semigroups (on white noise functionals), along with their fundamental properties: invariant measures, invariant white noise operators, and mean ergodicity. Then, utilizing canonical topological isomorphisms between the spaces of two-variable white noise functionals and the spaces of white noise operators, we formulate a quantum analogue of the Gaussian generalized Mehler semigroups and discuss their fundamental properties. By establishing the notion of the positivity of white noise operators, we introduce the concept of quantum dynamical (Markov) semigroups acting on the space of white noise operators. Finally, we discuss the Gaussianity of the quantum dynamical semigroups induced by the Gaussian generalized Mehler semigroups. The construction of the quantum dynamical semigroups leads to the general notion of quantum dynamical semigroups on generalized operators.
We consider random linear unbounded operators on a Banach space & Xscr;. For example, such random operators may be random quantum channels. The Law of Large Numbers is known when & Xscr; is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of the sum of i.i.d. variables, there may be considered the composition of random semigroups e(i)(A)t/n. We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of unbounded linear operators on a uniformly smooth Banach space.
We prove that the Kadison-Kastler and Christensen distances are stable under the Banach space injective tensor product (respectively, the Banach space projective tensor product) of a Banach space with any unital commutative C-& lowast;-algebra (respectively, of a C-& lowast;-algebra with any unital C-& lowast;-algebra). Apart from these stability results, we make some explicit calculations of the Kadison-Kastler, Christensen and Mashood-Taylor distances between certain subalgebras of some crossed-product operator algebras.
This paper studies regularity properties of multiplicative stochastic processes on infinite-dimensional Lie groups. We investigate conditions under which these processes admit c & agrave;dl & agrave;g modifications and derive bounds on their local behavior. Our approach builds on the local equivalence of Banach-Lie groups and Banach spaces via the exponential and logarithm, allowing us to transfer analytic estimates and structural results. To illustrate our findings, we consider multiplicative processes on the Heisenberg group.
We study the stochastic Burgers equation driven by an additive Hermite sheet of order q >= 1. The equation is formulated in the mild sense using the heat semigroup, and existence and uniqueness of solutions are established via a fixed-point argument in suitable Banach spaces. Under appropriate conditions on the Hurst parameters of the Hermite sheet, we derive uniform moment estimates for the solution, which form the basis for the regularity analysis. We prove that the solution admits a continuous modification that is H & ouml;lder continuous in both time and space, with exponents determined by the Hurst parameters of the driving noise. In addition, we show that the solution inherits an anisotropic self-similarity property from the Hermite sheet, and we identify the corresponding scaling exponents. The additive noise structure allows the stochastic convolution to be defined through multiple Wiener-It & ocirc; integrals with deterministic kernels. As a consequence, the analysis avoids Malliavin calculus techniques that are typically required for non-Gaussian noises of Hermite rank q >= 2.
In this paper, we investigate the rate of convergence toward the Boolean extreme value distribution, which is the universal limiting law for the normalized spectral maximum of Boolean independent and identically distributed positive operators, under the von Mises condition.
The approximative calculation of iterated nested expectations is a recurring challenging problem in applications. Nested expectations appear, for example, in the numerical approximation of solutions of backward stochastic differential equations (BSDEs), in the numerical approximation of solutions of semilinear parabolic partial differential equations (PDEs), in statistical physics, in optimal stopping problems such as the approximative pricing of American or Bermudan options, in risk measure estimation in mathematical finance, or in decision-making under uncertainty. Nested expectations which arise in the above named applications often consist of a large number of nestings. However, the computational effort of standard nested Monte Carlo approximations for iterated nested expectations grows exponentially in the number of nestings and it remained an open question whether it is possible to approximately calculate multiply iterated high-dimensional nested expectations in polynomial time. In this article we tackle this problem by proposing and studying a new class of full-history recursive multilevel Picard (MLP) approximation schemes for iterated nested expectations. In particular, we prove under suitable assumptions that these MLP approximation schemes can approximately calculate multiply iterated nested expectations with a computational effort growing at most polynomially in the number of nestings $ K \in \mathbb{N} = \{1, 2, 3, \ldots \} $, in the problem dimension $ d \in \mathbb{N} $, and in the reciprocal $\frac{1}{\varepsilon}$ of the desired approximation accuracy $ \varepsilon \in (0, \infty) $.
Generators of Gaussian QMS have a GKLS form with noise operators that are linear in creation and annihilation operators, and Hamiltonian which is quadratic. We characterize Gaussian QMS arising in the Weak Coupling Limit (WCL) of a system interacting with some reservoirs. It turns out that their generators, under some natural assumptions, have a GKLS representation with noise operators that are either annihilation or creation operators and Hamiltonian [Formula: see text] which is diagonalizable as a quadratic operator.
In this paper, we introduce and study a model of discrete-time quantum walk based on quantum Bernoulli noises, which can be viewed as a perturbation of a model existing in the literature. We first construct a collection of self-adjoint unitary operators via quantum Bernoulli noises. Then, with these operators as the shift operators, we introduce a model of discrete-time quantum walk. We examine basic properties of the model via its shift operators and establish a formula for calculating its probability distributions. Finally, we investigate stationary measures of the model, and explicitly obtain a family of such measures.
